A guide to the Choquard equation (Q523358)

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scientific article

Language	Label	Description	Also known as
English	A guide to the Choquard equation	scientific article

Statements

scholarly article

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A guide to the Choquard equation (English)

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Jean Van Schaftingen

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Journal of Fixed Point Theory and Applications

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publication date

20 April 2017

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full work available at URL

https://arxiv.org/abs/1606.02158

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https://cronfa.swan.ac.uk/Record/cronfa31722/Download/0031722-23012017182500.pdf

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zbMATH Keywords

Choquard equation

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Pekar polaron

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Schrödinger-Newton equation

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focusing Hartree equation

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attractive nonlocal interaction

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Riesz potential

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MaRDI profile type

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On a periodic Schrödinger equation with nonlocal superlinear part

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A nonlinear superposition principle and multibump solutions of periodic Schrödinger equations

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Existence and concentration of ground state solutions for a critical nonlocal Schrödinger equation in \(\mathbb R^2\)

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Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field

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Multi-bump solutions for Choquard equation with deepening potential well

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Existence of semiclassical ground state solutions for a generalized Choquard equation

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Multiplicity and concentration of solutions for a quasilinear Choquard equation

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On Schrödinger-Poisson systems

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Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \(\mathbb{R}^N\), and related topics

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MULTIPLE BOUND STATES FOR THE SCHRÖDINGER–POISSON PROBLEM

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Dual variational methods in critical point theory and applications

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A variational approach to the Schrödinger-Poisson system: asymptotic behaviour, breathers, and stability

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Symmetry breaking regime in the nonlinear Hartree equation

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Generalized Schrödinger-Poisson type systems

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The Schrödinger–Newton equation and its foundations

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Partial symmetry of least energy nodal solutions to some variational problems

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Existence of groundstates for a class of nonlinear Choquard equations in the plane

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Pitt's inequality with sharp convolution estimates

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Pitt's inequality and the fractional Laplacian: Sharp error estimates

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Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems

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Existence of positive solutions of the equation \(-\Delta u+a(x)u=u^{(N+2)/(N-2)}\) in \({\mathbb{R}}^ N\)

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An eigenvalue problem for the Schrödinger-Maxwell equations

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The Thomas-Fermi-von Weizsäcker theory of atoms and molecules

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Nonlinear scalar field equations. I: Existence of a ground state

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Soliton dynamics for the generalized Choquard equation

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Bound state solutions for a class of nonlinear Schrödinger equations

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A general rearrangement inequality for multiple integrals

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Remarks on the Schrödinger operator with singular complex potentials

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A Relation Between Pointwise Convergence of Functions and Convergence of Functionals

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Positive solutions of nonlinear elliptic equations involving critical sobolev exponents

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An approach to symmetrization via polarization

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Existence of a Nontrivial Solution to a Strongly Indefinite Semilinear Equation

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Cases of equality in the Riesz rearrangement inequality

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Multi-peak standing waves for nonlinear Schrödinger equations with a general nonlinearity

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Standing waves for nonlinear Schrödinger equations with a general nonlinearity

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Standing waves with a critical frequency for nonlinear Schrödinger equations. II.

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On the standing waves for nonlinear Hartree equation with confining potential

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A sign-changing solution for a superlinear Dirichlet problem

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Existence of steady states for the Maxwell-Schrödinger-Poisson system: exploring the applicability of the concentration-compactness principle

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On some periodic Hartree-type models for crystals

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Some existence results for superlinear elliptic boundary value problems involving critical exponents

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Blow up solutions for one class of system of Pekar-Choquard type nonlinear Schrödinger equation

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Existence of a nontrivial solution for a strongly indefinite periodic Choquard system

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Qualitative properties of solutions for an integral equation

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Classification of solutions for an integral equation

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Classification of isolated singularities of positive solutions for Choquard equations

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The one-dimensional Schrödinger-Newton equations

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Stationary solutions of the Schrödinger-Newton model -- an ODE approach.

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Multiple solutions to a magnetic nonlinear Choquard equation

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Intertwining semiclassical solutions to a Schrödinger-Newton system

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Multiple \(\mathbb {S}^{1}\)-orbits for the Schrödinger-Newton system.

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Ground states for the pseudo-relativistic Hartree equation with external potential

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Semi-classical limit for Schrödinger equations with magnetic field and Hartree-type nonlinearities

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On the planar Schrödinger-Poisson system

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Positive and sign changing solutions to a nonlinear Choquard equation

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Existence of ground states for nonlinear, pseudo-relativistic Schrödinger equations

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SOLITON DYNAMICS FOR THE SCHRÖDINGER–NEWTON SYSTEM

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On fractional Choquard equations

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Constraint minimizers of mass critical Hartree energy functionals: Existence and mass concentration

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Semi-classical states for nonlinear Schrödinger equations

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Elementary Proofs of Embedding Theorems for Potential Spaces of Radial Functions

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Semiclassical stationary states for nonlinear Schrödinger equations under a strong external magnetic field

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Some a priori estimates for a critical Schrodinger-Newton equation

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Asymptotics for the polaron

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Fractional integrals on radial functions with applications to weighted inequalities

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Some Theorems About the Riesz Fractional Integral

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Higher dimensional solitary waves generated by second-harmonic generation in quadratic media

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Sharp threshold of global existence and instability of standing wave for the Schrödinger-Hartree equation with a harmonic potential

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The ground state energy of a polaron in a strong magnetic field

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Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential

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Inversion positivity and the sharp Hardy-Littlewood-Sobolev inequality

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The Schrödinger–Newton system with self-field coupling

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Boson stars as solitary waves

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The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

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On perturbation of a functional with the mountain pass geometry: application to the nonlinear Schrödinger-Poisson equations and the nonlinear Klein-Gordon-Maxwell equations

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Soliton and blow-up solutions to the time-dependent Schrödinger-Hartree equation

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Pointwise bounds and blow-up for Choquard-Pekar inequalities at an isolated singularity

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Least action nodal solutions for the quadratic Choquard equation

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Nodal solutions for the Choquard equation

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Self-dual Partial Differential Systems and Their Variational Principles

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Symmetry and related properties via the maximum principle

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On a class of non linear Schrödinger equations with non local interactions

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The Schrödinger–Newton equation as a non-relativistic limit of self-gravitating Klein–Gordon and Dirac fields

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On the magnetic Pekar functional and the existence of bipolarons

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Instability of standing waves for inhomogeneous Hartree equations

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Schrödinger systems arising in nonlinear optics and quantum mechanics. I

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On Schrödinger systems arising in nonlinear optics and quantum mechanics: II.

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Spectral theory of the operator \((p^2+m^2)^{1/2}-Ze^2/r\)

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Nonlinear scalar field equations in \(\mathbb{R}^N\): Mountain pass and symmetric mountain pass approaches

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GROUND AND BOUND STATES FOR A STATIC SCHRÖDINGER–POISSON–SLATER PROBLEM

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Existence of solutions with prescribed norm for semilinear elliptic equations

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Quasiclassical soliton solutions of the Hartree equation. Newtonian interaction with screening

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Rearrangements and convexity of level sets in PDE

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MULTIPLE POSITIVE SOLUTIONS AND BIFURCATION FOR AN EQUATION RELATED TO CHOQUARD’S EQUATION

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From atoms to crystals: a mathematical journey

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On the regularity of positive solutions of a class of Choquard type equations

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Qualitative Analysis for the Static Hartree-Type Equations

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Uniqueness of ground states for pseudorelativistic Hartree equations

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Derivation of Hartree's theory for generic mean-field Bose systems

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The Chandrasekhar theory of stellar collapse as the limit quantum mechanics

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Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation

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Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities

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Exact ground state energy of the strong-coupling polaron

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The Choquard equation and related questions

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The concentration-compactness principle in the calculus of variations. The locally compact case. I

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Solutions of Hartree-Fock equations for Coulomb systems

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Symmetry of minimizers for some nonlocal variational problems

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Existence and concentration of solutions for a nonlinear Choquard equation

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Stationary solutions for the non-linear Hartree equation with a slowly varying potential

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Classification of positive solitary solutions of the nonlinear Choquard equation

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An example of stable excited state on nonlinear Schrödinger equation with nonlocal nonlinearity.

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The Schrödinger-Newton equations beyond Newton

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Critical point theory and Hamiltonian systems

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On regular solutions of a nonlinear equation of Choquard's type

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Groundstates and radial solutions to nonlinear Schrödinger-Poisson-Slater equations at the critical frequency

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Spherically-symmetric solutions of the Schrödinger-Newton equations

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Semiclassical stationary states for nonlinear Schrödinger equations with fast decaying potentials

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Nonlocal Hardy type inequalities with optimal constants and remainder terms

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Nonexistence and optimal decay of supersolutions to Choquard equations in exterior domains

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Groundstates of nonlinear Choquard equations: existence, qualitative properties and decay asymptotics

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Semi-classical states for the Choquard equation

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Existence of groundstates for a class of nonlinear Choquard equations

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Groundstates of nonlinear Choquard equations: Hardy–Littlewood–Sobolev critical exponent

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Weighted Norm Inequalities for Fractional Integrals

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The Schrödinger–Poisson System with Positive Potential

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Pseudorelativistic Hartree Equation with General Nonlinearity: Existence, Non-existence and Variational Identities

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Breathing modes for the Schrödinger-Poisson system with a multiple-well external potential

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On gravity's role in quantum state reduction

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Quantum computation, entanglement and state reduction

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A ground state alternative for singular Schrödinger operators

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Isoperimetric Inequalities in Mathematical Physics. (AM-27)

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On a class of nonlinear Schrödinger equations

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L'intégrale de Riemann-Liouville et le problème de Cauchy pour l'équation des ondes

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L'intégrale de Riemann-Liouville et le problème de Cauchy

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On quantization of fields

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One-dimensional representation, inversion, and certain properties of the Riesz potentials of radial functions

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The Schrödinger-Poisson equation under the effect of a nonlinear local term

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On the Schrödinger-Poisson-Slater system: behavior of minimizers, radial and nonradial cases

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Odd symmetry of least energy nodal solutions for the Choquard equation

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Vortex-type solutions to a magnetic nonlinear Choquard equation

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General relativistic boson stars

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A note on Schrödinger-Newton systems with decaying electric potential

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On a theorem of functional analysis

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Boundedness of translation invariant operators on Hölder spaces and \(L^ p\)-spaces

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Variational Methods

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Bifurcation for variational problems when the linearisation has no eigenvalues

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Bound states of the Schrödinger-Newton model in low dimensions

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On nonhomogeneous elliptic equations involving critical Sobolev exponent

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The ground state energy of the Schrödinger-Newton equation

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Asymptotics and inversion of Riesz potentials through decomposition in radial and spherical parts

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An analytical approach to the Schrödinger-Newton equations

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Ground states for Schrödinger-Poisson type systems

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Existence of Bound States for Schrödinger-Newton Type Systems

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Symmetry of solutions of semilinear elliptic problems

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Choquard equations under confining external potentials

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Minimax theorems

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Uniqueness and nondegeneracy of ground states for Choquard equations in three dimensions

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Existence of multiple positive solutions for Choquard equation with perturbation

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Sharp constants in the Hardy-Rellich inequalities

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Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part

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Existence and multiplicity of solutions for nonlinear Schrödinger equations with magnetic field and Hartree type nonlinearities

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Multi-peak solutions for nonlinear Choquard equation with a general nonlinearity

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The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in \(\mathbb R^N\)

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Multiple solutions of nonhomogeneous Chouquard's equations

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Existence of a nontrivial solution for Choquard's equation

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On the existence of solutions for the Schrödinger-Poisson equations

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Identifiers

zbMATH Open document ID

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10.1007/s11784-016-0373-1

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Mathematics Subject Classification ID

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zbMATH DE Number

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Sitelinks

Mathematics(1 entry)

mardi Publication:523358

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